Alright, even though intuitively this looks like something that should work, it doesn't. This is because of the way _Mathematica_ handles precision. You might have noticed that you can output really large numbers like *10^10000* which are way beyond `MachinePrecision` of `double`. This is because _Mathematica_ is a symbolic language. Each value gets it's precision which you can look at by calling ```Precision[a]``` Symbols, Integers and some other stuff has `MachinePrecision`. It is written in the documentation that

> N[e] typically works by replacing numbers with machine numbers and computing the result. 

and that: 

> The precision of the result is the same as that of the input: 

This means if you have machine precision it will `N` will always return the same precision. In your sanity check if you wrote 

`N[5.,10]` instead of `N[5,10]` you would get `5.` 

Now you can change the precision by using `SetPrecision` as I did in the comment and things will go smoother. 

**EDIT** 

Also `N` Will try to give you the result that can be *Exactly* computed without machine errors. So there's also that.